rational functions
rational functions
Oblique Asymptotes
MHF4U: Advanced Functions
In addition to horizontal and vertical asymptotes, a function
may have oblique asymptotes.
Oblique asymptotes are sometimes called “slant” asymptotes
because they have the form y = ax + b, where a 6= 0.
Oblique Asymptotes
A function will have an oblique asymptote if the degree of
the numerator is one greater than that of the denominator.
J. Garvin
A function will never have both oblique and horizontal
asymptotes.
J. Garvin — Oblique Asymptotes
Slide 2/16
Slide 1/16
rational functions
rational functions
Oblique Asymptotes
Oblique Asymptotes
Example
There is a vertical asymptote at x = 0, as given by the
denominator.
Determine the equation of the oblique asymptote for
x2 − 1
f (x) =
, and graph the function.
x
Use long division to determine the equation of the oblique
asymptote.
x
There is no f (x)-intercept, since setting x = 0 causes a
division by zero error.
Since x 2 − 1 is a difference of squares, f (x) has x-intercepts
at ±1.
These points, along with the asymptotes, give us enough
information to accurately sketch the graph, but we can test
values of x close to the vertical asymptote to get a better
picture.
x
x2 − 1
− x2
As x → 0 from the left, f (x) → ∞, and as x → 0 from the
right, f (x) → −∞.
The equation of the oblique asymptote is y = x.
J. Garvin — Oblique Asymptotes
Slide 4/16
J. Garvin — Oblique Asymptotes
Slide 3/16
rational functions
Oblique Asymptotes
rational functions
Oblique Asymptotes
Example
Graph f (x) =
x 2 + 2x − 3
.
x +1
x +1
x +1
x 2 + 2x − 3
− x2 − x
x −3
−x −1
−4
There is an oblique asymptote with equation y = x + 1.
J. Garvin — Oblique Asymptotes
Slide 5/16
J. Garvin — Oblique Asymptotes
Slide 6/16
rational functions
Oblique Asymptotes
rational functions
Oblique Asymptotes
There is a vertical asymptote at x = −1, since the
denominator is zero when x = −1.
Setting x = 0 gives an f (x)-intercept of −3.
Since f (x) factors as f (x) =
x-intercepts at 1 and −3.
(x − 1)(x + 3)
, there are
x +1
As x → 0− , f (x) → ∞, and as x → 0+ , f (x) → −∞.
In the notation above, x → k − means “as x approaches k
from the left”, while x → k + is from the right.
J. Garvin — Oblique Asymptotes
Slide 7/16
J. Garvin — Oblique Asymptotes
Slide 8/16
rational functions
Oblique Asymptotes
rational functions
Oblique Asymptotes
Example
Graph f (x) =
x 2 + 2x
.
x +2
x(x + 2)
= x, where x 6= −2.
x +2
Thus, the graph of f (x) is the same as the graph of y = x,
but with a point discontinuity at x = −2.
In this case, note that f (x) =
It is generally a good idea to check for the same root in both
the numerator and denominator before doing any extra work.
J. Garvin — Oblique Asymptotes
Slide 9/16
J. Garvin — Oblique Asymptotes
Slide 10/16
rational functions
rational functions
Oblique Asymptotes
Oblique Asymptotes
A More Complex Example
Use long division to determine the equation of the oblique
asymptote.
x 3 − 2x 2 − x + 2
Graph f (x) =
.
x2 − x − 6
g (x)
Let f (x) =
.
h(x)
By the FT, g (1) = g (2) = g (−1) = 0, and
h(−2) = h(3) = 0.
(x − 1)(x − 2)(x + 1)
.
(x + 2)(x − 3)
There are vertical asymptotes at x = −2 and x = 3.
x2 − x − 6
− x 2 + 5x + 2
x2 − x − 6
4x − 4
Therefore, f (x) =
There is an oblique asymptote with equation y = x − 1.
The x-intercepts are at 1, 2 and −1.
The f (x)-intercept is at − 31 .
J. Garvin — Oblique Asymptotes
Slide 11/16
x −1
x 3 − 2x 2 − x + 2
− x 3 + x 2 + 6x
J. Garvin — Oblique Asymptotes
Slide 12/16
rational functions
rational functions
Oblique Asymptotes
Oblique Asymptotes
Putting everything together produces the following graph.
Test points on either side of the vertical asymptotes.
As x → −2− , f (x) → −∞, so the function decreases.
As x → −2+ , f (x) → ∞, so the function increases.
As x → 3− , f (x) → −∞, so the function decreases.
As x → 3+ , f (x) → ∞, so the function increases.
This gives us enough information to make a rough sketch.
Clearly, more information is needed for an accurate graph.
J. Garvin — Oblique Asymptotes
Slide 13/16
J. Garvin — Oblique Asymptotes
Slide 14/16
rational functions
Oblique Asymptotes
rational functions
Questions?
Note that f (x) intersects the oblique asymptote at x = 1.
Since asymptotes describe end behaviour of a function, this is
perfectly normal.
J. Garvin — Oblique Asymptotes
Slide 15/16
J. Garvin — Oblique Asymptotes
Slide 16/16